Geodesic learning

Learning is a fundamental characteristic of living systems, enabling them to comprehend their environments and make informed decisions. These decision-making processes are inherently influenced by available information about their surroundings and specific objectives. There is an intriguing perspective is that each process is highly efficient under a given set of conditions. A key question, then, is how close to optimality it is or how efficient it is under given conditions. Here, the concept of geodesic learning as the optimal reference process, with which each process can be compared, is introduced and formulated on the basis of geometry. The probability distribution describing the state of the composite system consisting of the environment, termed the information bath, and a decision-maker is described by use of the entropic quantities. This enables one to study the system in analogy with thermodynamics. Learning processes are expressed as the changes of parameters contained in the distribution. For a geometric interpretation of the processes, the manifold endowed with the Fisher-Rao metric as the Riemannian metric is considered. This framework allows one to conceptualize the optimality of each process as a state change along a geodesic curve on the manifold, which gives rise to geodesic learning. Then, the bivariate Gaussian model is presented, and the processes of geodesic learning and adaptation are analyzed for illustrating this approach.